Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.

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1answer
49 views

What does being diffeomorphic mean in the context of configuration spaces?

A sphere space can serve as a "model space" for any configuration space that is diffeomorphic to the sphere space. This is a quote from my text book (Principles of Robot Motion: Theory, ...
2
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0answers
37 views

Name for classes of algebras closed under products and quotients

A class of algebras closed under products, quotetiens and subalgebras is a variety. Is there a name for a class of algebras closed under products and quotients? Could you refer me to any theorems ...
1
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0answers
67 views

The word “onto” - adjective?

According to Oxford English dictionary, the word "onto" is preposition only but I see it used as an adjective in mathematical writings. I think it is grammatically correct to say that "$f$ maps $X$ ...
3
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1answer
53 views

Is there a standard notation for building sets up form a given one?

In ZFC each set $S$ has a well-founded membership tree building $S$ up from the empty set $\emptyset$. You could attach the membership tree for any given set $A$ on each of the bottom nodes for the ...
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1answer
92 views

Where is the border between functional analysis and real analysis?

I always thought that real analysis deals with analysis on the real line, eventually on the Euclidean space $\mathbb R^n$. But why does someone have to label a course as real analysis when it is ...
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0answers
44 views

Cohomology calculation or computation?

I have a terminology question: Does one compute the cohomology of a group, or does one calculate it? Is it more common to speak of cohomology calculation or cohomology computation? Thanks for your ...
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0answers
27 views

Space of functions in the upper half-plane

Let $f(\tau)$ be a (say, holomorphic) function in the upper half-plane. Consider $$ A={\rm Span}_{\mathbb{C}} \{ f(\gamma \cdot \tau) : \gamma \in SL_2 (\mathbb{Z}) \}. $$ Is there a standard name ...
6
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0answers
97 views

Is there an accepted term for those objects of a category $X$ such that for all $Y$, there is at most one arrow $X \rightarrow Y$?

In category theory, I have seen "weakly initial object" used as follows: $X$ is weakly initial iff for all objects $Y,$ there is at least one arrow $X \rightarrow Y$. Of course, another way of ...
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1answer
52 views

What is the terminology for “lemma of lemma”

Let's say I need to prove a main theorem, to prove which I need three lemmas. Thus in writing the structure is as follows: Lemma 1 Proof Lemma 2 Proof Lemma 3 Proof ...
3
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1answer
187 views

Do you decline a multiplier in reading a mathematical formula in Russian?

How do you read "Порядок определителя равен $2n$"? Is it "двум эн" or is it "два эн"? And in a sum, do you read $c = a_5 + a_6$ as "це равно а пятому плюс а шестому"? Or does the plus sign interfere ...
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3answers
64 views

Which quadrant is the “first quadrant”?

In the coordinate plane split into four quadrants by the $x$- and $y$-axes, I learned (educated in a public school in the U.S.) that the "first quadrant" was the one with both $x$ and $y$ positive, ...
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1answer
50 views

Notation for the number of times one element divides another.

Let $R$ denote a commutative ring with unity. Consider elements $a,b \in R$. Is there an accepted notation (like $a \| b$ or some such) for the number of times that $a$ divides $b$? Explicitly, we can ...
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0answers
83 views

Reading mathematical formulas in Russian & German

The book Russian for Mathematicians by Glazunova has a very useful section with examples of how formulas are read in Russian. (Most mathematical dictionaries don't seem to have this, as I suppose they ...
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3answers
79 views

Elementary “binomial theorem” in English

In German schools, the identitiy $(a+b)^2 = a^2 + 2ab + b^2$ is called the first binomial formula (literally translated). However, it seems to me this English term only refers to the more general ...
5
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1answer
150 views

Name for Theorem 3.27 from baby Rudin?

Rudin rarely gives names to the theorems in this book. Theorem 3.27 states if $\{a_n\}$ is a monotonically decreasing sequence of positive reals, then $$\sum_{n=1}^\infty a_n\,\text{ ...
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1answer
60 views

What is the difference between helix and spiral? [closed]

The words spiral and helix are both used for curves that "wind around". For example, both searches "DNA spiral" and "DNA helix" (with quotation marks) result in many thousands of Google hits. Is ...
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0answers
33 views

Mathematical principles and therems - difference?

For me, intuitively, a mathematical principle is simply an influential theorem. Still, I am not clear on how and who decides (or decided) if a theorem/statement is a mathematical principle. Can you ...
0
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1answer
19 views

Confusion about the definition of upper bounds of a set

I am confused about upper bounds of a set. Consider a set: $A = ${$1, 2, 3, 4, 5, 6, 7$} How many upperbounds are there? Does the upperbound need to be in the set? Also about supremum. What is ...
3
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1answer
43 views

“isometric invariant” vs “isometric” what do these term mean?

I am now hopelessly confused: There is Hilberts Theorem https://en.wikipedia.org/wiki/Hilbert%27s_theorem_%28differential_geometry%29 . that implies that there are no isometric embeddings of the ...
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0answers
13 views

Clarify what vector quantization is.

In the Wikipedia page for vector quantization the following definition is given (slightly paraphrased): A quantization technique which divides a large set of vectors into groups having ...
2
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3answers
310 views

Why are “imaginary numbers” imaginary?

I understand the definition that each member of $\Bbb{C}$ is of the form $a+bi$. I also recognize that $i^2 = -1$. What about this is imaginary? Like it has become a "real" number in the sense that we ...
5
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1answer
58 views

Scott continuity on powerset

I am looking for the name of the class of functions $f:\mathcal P(A)→\mathcal P(A)$ that are monotone and that are characterised by their image on finite subsets, i.e. the functions $f$ satisfying the ...
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0answers
25 views

What does “to first order in exponent” mean?

I am studying information theory on "Elements of Invormation theory" (Cover Thomas). I cannot understand the meaning of "to first order in exponent" in the following theorem: ...
5
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6answers
372 views

Can two shapes occupy the exact same area on a plane?

Suppose there are two triangles on a plane. The coordinates for each point of both triangles are the same. It seems to me that there is nothing to differentiate these two triangles, and as such, ...
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2answers
49 views

What is the difference between a function and a formula?

I think that the difference is that the domain and codomain are part of a function definition, whereas a formula is just a relationship between variables, with no particular input set specified. ...
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0answers
33 views

What is “dependence in a simple fashion”?

I'm reading Silverman/Tate's Rational Points on Elliptic Curves and pg 15 states: (1) $\quad aX^2 + bY^2 = cZ^2$ (to be solved in integers) "Legendre's theorem states that there is an integer m, ...
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1answer
39 views

Is there an accepted name for algebraic structures like $\mathbb{Q}_{>0}$ and $\mathbb{R}_{>0}$?

Question. Is there an accepted name for algebraic structures that, like $\mathbb{Q}_{>0}$ and $\mathbb{R}_{>0}$, are models of the algebraic theory presented as follows? Sorts: $U$ Functions: ...
3
votes
3answers
195 views

How to divide natural number N into M nearly equal summands?

How to divide natural number N into M nearly equal summands? For example, to divide 20 by 13, in geometric representation, I should get How to generate the sequence above? What is the name of ...
23
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4answers
1k views

Why is “mathematical induction” called “mathematical”?

One of my whims is that I never write "mathematical induction" but just "induction". We are doing maths, so what is the point about precising? We don't say "Let $f$ be a mathematical function from the ...
2
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3answers
65 views

Name of an axiom

What's the name of the following axiom: Let $A$ and $B$ be sets of real numbers, and $(\forall a\in A\text{ and }\forall b\in B)a \leq b$. Then $\exists c\in\mathbb R$ such that $(\forall a\in A\text{ ...
8
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1answer
67 views

A function that crosses each horizontal line only finitely many times

Consider a real function $f(x)$ (not necessarily continuous) defined on a finite interval. Given a constant $C$, divide the interval to sub-intervals such that, in every sub-interval, either ...
0
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1answer
28 views

Is there a mathematical graph to describe completing a collection?

Example: there are 10 items in a collection. Getting items in the collection when you start out with none seems fairly simple, with some duplicates being acquired over time. But when you get down to ...
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0answers
24 views

Terminology: how do people call the “normal generating set”?

Let $G$ be a group and let $x_1,\cdots,x_n\in G$ and let $A$ be the normal closure of $\{x_1,\cdots,x_n\}$; that is, the smallest (by inclusion) normal subgroup containing $\{x_1,\cdots,x_n\}$. Notice ...
0
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1answer
31 views

Can totality be defined in terms of left-totality (or right-totality)?

In the context of relations, can totality be defined in terms of left-totality (or right-totality)? I ask this because both properties have the word "totality" in their names and one may think that ...
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2answers
38 views

Is there any standard terminology for this property?

Let $f$ be a map whose domain is $X$. If $f$ satisfies the property that for all $x\in X$, $$f(f(x))=f(x)\text{,}$$ is there any standard name for such a function? Not sure if "projection" is the ...
1
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1answer
35 views

What does “exponent 2 nilpotency class 2” mean?

According to the book The Symmetries of Things, p. 208, the number of groups of order 2048 "strictly exceeds 1,774,274,116,992,170, which is the exact number of groups of order 2048 that have ...
1
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1answer
55 views

Is there a name for this result in planar geometry?

I found out that the following statement is fairly easy to prove: Let $A$, $B$ and $C$ be thee distinct points in the plane. Let $S_{AB}$ be the circle that has the line segment $AB$ as a ...
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1answer
78 views

How to describe the Cartesian product $\mathbb{R} × \mathbb{R}$?

I am taking a discrete mathematics course in the spring and in an attempt to fully understand the material I am reading ahead. I came across this statement Let $\mathbb{R}$ denote the set of all real ...
3
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1answer
34 views

Is there any significance to complex function “monotone in norm?”

So, I was reading a question earlier where someone asked if something would be strictly monotone in the complex plane, and the comment was that this would be meaningless, since the complex numbers ...
3
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2answers
47 views

Definition of totality in relations

I see two apparently different definitions for totality which don't seem to be equivalent. Definition 1. A relation $R \subset X \times Y$ is total if it associates to every $x \in X$ at least one $y ...
3
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2answers
80 views

What is a non-decreasing sequence of sets?

What is a non-decreasing sequence of sets and how come it can have a limit? It appear in a probability theory book
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2answers
50 views

Is there a good short phrase for a point where a function is continuous but not smooth?

Given a point $x_0$ where a function $f$ is $C^0$ but not $C^1$, how could one call this point intuitively? I am not looking for a technically precise term (like a point where $f'$ is ...
3
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1answer
79 views

What does “versin” mean?

$$\newcommand{\versin}{\operatorname{versin}}2\versin A+\cos ^2 A= 1+\versin ^2 A$$ I don't understand the word 'ver' in this equation. What does it mean?
3
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0answers
25 views

How is a part of eulerian path called?

An eulerian path in a graph is a path that visits every edge in the graph exactly once. If there is a path that has a similar property that it visits an edge at most once (e.g. a part of an eulerian ...
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1answer
58 views

What format is this?

I was given a snippet and can't seem to parse it myself, what's the name of this format and is there a tool that will render it like latex or mathML like this site does? ...
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2answers
75 views

Opposite of a function being bijective?

A function is bijective if it is both surjective and injective. Is there a term for when a function is both not surjective and not injective?
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0answers
39 views

Name for matrices with $a_{ij} + a_{ji} = 1$?

Do you know of any commonly used name for square matrices $A$ having the property that $$ a_{ij} + a_{ji} = 1$$ for all $i,j \in \{1,\dots, n\}$, where $n$ is the dimension of $A$?
2
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1answer
48 views

Can there be a bijection between a countably infinite set and an uncountably infinite set?

I suppose the answer is trivially no, but I haven't actually seen it stated precisely this way in the general. I've only seen specific cases, such as the proof by Cantor's diagonal argument that ...
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0answers
56 views

Is it normal (correct) to calculate a probability without knowing the sample space?

Is it normal (correct) to calculate a probability without knowing the sample space? Background: I have finished a probability calculation $\mathbb{P}(E)$. I want to do some simulations. ...
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0answers
52 views

Generalized semilattice morphism

Join-semilattice morphism from a join-semilattice $\mathfrak{A}$ to a join-semilattice $\mathfrak{B}$ is a function $\alpha$ conforming to the formula $\alpha(X\sqcup Y) = \alpha X\sqcup\alpha Y$ ...